17. Periodic Motion

Simple Harmonic Motion of Pendulums

Open Question
(II) A musician wishes to construct a metronome based on a simple pendulum design where a “beat” is sounded when the bob reaches each opposite extreme of its swing. How long must the pendulum be so that this metronome is set for 104 bpm (bpm = beats per minute)?
Open Question
Interestingly, there have been several studies using cadavers to determine the moments of inertia of human body parts, information that is important in biomechanics. In one study, the center of mass of a 5.0 kg lower leg was found to be 18 cm from the knee. When the leg was allowed to pivot at the knee and swing freely as a pendulum, the oscillation frequency was 1.6 Hz. What was the moment of inertia of the lower leg about the knee joint?
Open Question
A uniform rod of length L oscillates as a pendulum about a pivot that is a distance x from the center.a. For what value of x, in terms of L, is the oscillation period a minimum?
Open Question
A 15-cm-long, 200 g rod is pivoted at one end. A 20 g ball of clay is stuck on the other end. What is the period if the rod and clay swing as a pendulum?
Open Question
A uniform rod of mass M and length L swings as a pendulum on a pivot at distance L/4 from one end of the rod. Find an expression for the frequency f of small-angle oscillations.
Open Question
Astronauts on the first trip to Mars take along a pendulum that has a period on earth of 1.50 s. The period on Mars turns out to be 2.45 s. What is the free-fall acceleration on Mars?
Open Question
A pendulum on a 75-cm-long string has a maximum speed of 0.25 m/s. What is the pendulum's maximum angle in degrees?
Open Question
A pendulum is made by tying a 75 g ball to a 130-cm-long string. The ball is pulled 5.0° to the side and released. How many times does the ball pass through the lowest point of its arc in 7.5 s?
Open Question
In a science museum, a 110 kg brass pendulum bob swings at the end of a 15.0-m-long wire. The pendulum is started at exactly 8:00 a.m. every morning by pulling it 1.5 m to the side and releasing it. Because of its compact shape and smooth surface, the pendulum's damping constant is only 0.010 kg/s. At exactly 12:00 noon, how many oscillations will the pendulum have completed and what is its amplitude?
Open Question
(II) What is the period of a simple pendulum 47 cm long
(b) when it is in a freely falling elevator?
Open Question
A simple pendulum oscillates with frequency ƒ What is its frequency if the entire pendulum accelerates at 0.35 g

(b) downward?
Open Question
(II) The balance wheel of a mechanical (analog) watch is a thin ring of radius 0.95 cm attached to a coil spring (roughly like Fig. 14–35). It oscillates with a frequency of 1.55 Hz. If a torque of 1.1 x 10⁻⁵ m • N causes the wheel to rotate 45°, calculate the mass of the balance wheel.
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Open Question
(II) A plywood disk of radius 20.0 cm and mass 2.20 kg has a small hole drilled through it, 2.00 cm from its edge (Fig. 14–38). The disk is hung from the wall by means of a metal pin through the hole, and is used as a pendulum. What is the period of this pendulum for small oscillations?
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Open Question
Tall buildings are designed to sway in the wind. In a 100 km/h wind, suppose the top of a 110-story building oscillates horizontally with an amplitude of 15 cm at its natural frequency, which corresponds to a period of 7.0 s. Assuming SHM, find the maximum horizontal velocity and acceleration experienced by an employee as she sits working at her desk located on the top floor. Compare the maximum acceleration (as a percentage) with the acceleration due to gravity.
Open Question
A “seconds” pendulum has a period of exactly 2.000 s. That is, each one-way swing takes 1.000 s. What is the length of a seconds pendulum in Austin, Texas, where g = 9.793 m /s² ? If the pendulum is moved to Paris, where g = 9.809 m/s², by how many millimeters must we lengthen the pendulum? What is the length of a seconds pendulum on the Moon, where g = 1.62 m/s² ?